Calculation of the geometric and kinematic characteristics of the engagement of the arched teeth of a cylindrical gear with errors in the relative position

The use of cylindrical gears with arched teeth in the drives of energy-saturated machines (locomotives, final drives of tractors, top drives of drilling rigs, helicopters) makes it possible to multiply the durability of gears compared to spur and helical gears. To date, a dozen and a half different techniques and methods for designing and manufacturing arched teeth of cylindrical wheels have been proposed. Among them, the most promising is the method of cutting arched teeth on CNC machines with circular cutter heads using the running-in method with a single division. The purpose of the article is to implement the construction of mathematical models for the formation of arched teeth on the basis of the theory of spatial gearing and to study the geometric and kinematic characteristics of their gearing under conditions of non-zero errors in the relative position of the wheel and gear in the transmission. The object of study is a cylindrical gear with arched teeth. In the article, dependencies for radius-vectors and unit vectors of the normals of the surfaces of arched teeth of cylindrical wheels are obtained, a mathematical model is developed for calculating the points of contact of arched teeth in engagement in the presence of a misalignment angle, a non-parallelism angle and wheel displacement. On the example of the final transmission of a locomotive, for the first time, the influence of the density of fit of the surfaces of the teeth in engagement on the displacement of the working lines from the average section of the tooth due to errors in the position of the transmission elements was studied for the first time. By calculating the error of the position function in the transmission, the procedure for quantifying its nonconjugacy is considered. It has been established that the geometry of the arched teeth of the method under study, under conditions of deviation of the positions of the transmission elements from the nominal ones, leads to a mismatch of the engagement, the position error function has a sawtooth shape, as a result of which impacts are inevitable when the teeth enter the engagement. Ways are proposed to achieve, within the framework of the considered method of manufacturing arched teeth, an approximate nature of the engagement of the teeth in the gear.